Solution of the pulse width modulation problem using orthogonal polynomials and Korteweg-de Vries equations

The mathematical underpinning of the pulse width modulation (PWM) technique lies in the attempt to represent “accurately” harmonic waveforms using only square forms of a fixed height. The accuracy can be measured using many norms, but the quality of the approximation of the analog signal (a harmonic form) by a digital one (simple pulses of a fixed high voltage level) requires the elimination of high order harmonics in the error term. The most important practical problem is in “accurate” reproduction of sine-wave using the same number of pulses as the number of high harmonics eliminated. We describe in this paper a complete solution of the PWM problem using Padé approximations, orthogonal polynomials, and solitons. The main result of the paper is the characterization of discrete pulses answering the general PWM problem in terms of the manifold of all rational solutions to Korteweg-de Vries equations.

Patch-clamp and amperometric recordings from norepinephrine transporters: Channel activity and voltage-dependent uptake

Transporters for the biogenic amines dopamine, norepinephrine, epinephrine and serotonin are largely responsible for transmitter inactivation after release. They also serve as high-affinity targets for a number of clinically relevant psychoactive agents, including antidepressants, cocaine, and amphetamines. Despite their prominent role in neurotransmitter inactivation and drug responses, we lack a clear understanding of the permeation pathway or regulation mechanisms at the single transporter level. The resolution of radiotracer-based flux techniques limits the opportunities to dissect these problems. Here we combine patch-clamp recording techniques with microamperometry to record the transporter-mediated flux of norepinephrine across isolated membrane patches. These data reveal voltage-dependent norepinephrine flux that correlates temporally with antidepressant-sensitive transporter currents in the same patch. Furthermore, we resolve unitary flux events linked with bursts of transporter channel openings. These findings indicate that norepinephrine transporters are capable of transporting neurotransmitter across the membrane in discrete shots containing hundreds of molecules. Amperometry is used widely to study neurotransmitter distribution and kinetics in the nervous system and to detect transmitter release during vesicular exocytosis. Of interest regarding the present application is the use of amperometry on inside-out patches with synchronous recording of flux and current. Thus, our results further demonstrate a powerful method to assess transporter function and regulation.